Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #199824 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,7,5,8,4,3)(9,10,13,18,15,17,14,23)(11,19,20,24,12,21,22,16);(1,3,6,8,5,7,4,2)(9,11,15,12)(10,16,17,24)(13,22,14,20)(18,21,23,19);(1,4,5,6)(2,7,8,3)(9,12,15,11)(10,16,17,24)(13,22,14,20)(18,19,23,21);(1,4,5,6)(2,7,8,3)(10,17)(16,24)(18,23)(19,21)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=h,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,f^c=fhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107172);